Another difficulty that lies in higher dimensions is the absence of a Beurling-Lax-Halmos type result for shift-invariant closed subspaces, which plays an important role in proving Proposition REF . Thus, it is evident that we require new techniques for proving an analogous result that can be applied to higher dimensions. Here, we follow an approach popularized by the methods of Halmos (in [1]}) and Shimorin (in [2]}) by focussing on abstract tuples of operators on Hilbert spaces that behave like shift operators on several rkHs's. More precisely, we are able to extend Proposition REF  to pure contractions commuting with certain tuples of left-invertible operators.
